Friday, February 28, 2003

I'm writing from the STACS conference, one of my favorite European conferences. Next week I'll be in Holland visiting CWI
and talking at the NVTI Theory Day.

Far less Americans than usual this year. While European travel is definitely down (my plane was quite empty), scientists are usually not deterred by political events. More likely is the drop in grant money forcing
researchers to be more picky in their choice of conferences. This year the Complexity Conference is in Europe as well.

Also far less French than usual, STACS being a joint French-German conference. I have no explanation for this.

This is not a science item but interesting nevertheless: I had to rebook my flight back; the US government needed the plane to "fight terrorism", most likely to ferry troops to the Gulf.

There have been some interesting talks at the conference. I'll try to mention some of them when I get a chance.

Tuesday, February 25, 2003

Russell Impagliazzo and Phillipe Moser have an upcoming Complexity paper
entitled A Zero-One Law for RP. What does that mean?

Jack Lutz defined a notion of resource-bounded measure to find the
relative size of complexity classes within exponential time. He has many
surveys and research papers available on
this interesting topic. Whether NP has measure zero in EXP is the most
exciting open question in this area.

Dieter van Melkebeek noted
that the Impagliazzo-Wigderson derandomization results implied a
zero-one law for BPP: Either
BPP=EXP and has measure one or there is sufficient derandomization to
show BPP has measure zero.

The ideas do not directly work for the one-sided error class RP. The
new Impagliazzo-Moser paper shows how to overcome these difficulties
to get the zero-one law for RP. They also show that if RP does not
have measure zero then not only RP but ZPP is
equal to EXP. They also show some new derandomization if NP does not
have measure zero, i.e., to get that AM = NP.

Friday, February 21, 2003

Today I saw Maria Chudnovsky give a talk at Princeton on her new result with
Paul Seymour finding a polynomial-time algorithm for testing perfect
graphs, a result I mentioned
in an earlier post. The algorithm actually tests for Berge graphs
which are graphs with no induced odd cycle of length at least five or
the complement of such a graph. An earlier result of Chudnovsky,
Robertson, Seymour and Thomas showed that a graph is perfect if and
only if it is Berge. Otherwise the new algorithm does not use the
techniques of the earlier paper.

The algorithm looks for some basic structures that would imply the
graph is not Berge. If these structures don't exist then one does a
cleaning process that simplifies the search for odd cycles. A
cleaning process is described in another paper by Chudnovsky,
Cornuéjols, Liu, Seymour and Vuškovic.

While the algorithm will test whether the graph is Berge, it is still
open to determine whether a graph had an odd cycle of length at least
five.

Chudnovsky's web
page now has papers describing all of these results as well as other
pointers on perfect graphs.

Monday, February 17, 2003

Let us call a nondeterministic Turing machine M balanced if for
every input x, all of its computational paths have the same length,
i.e., the number of nondeterministic bits depends only on x and not on
previous guesses. We can define the characterize class BPP as follows:

L is in BPP if there is a balanced nondeterministic polynomial-time M such that

If x is in L then there are at least twice as many accepting as
rejecting paths of M(x).

If x is not in L then there are at least twice as many rejecting
as accepting paths of M(x).

Suppose we use the same definition without the "balanced"
requirement. This gives us the class BPPpath studied mostly
in a 1997
paper of Han, Hemaspaandra and Thierauf.

BPPpath contains BPP by definition but it contains much
more. Let us show that SAT is contained in BPPpath.

Let φ be a formula with n variables. Consider the following
machine M(φ): Guess an assignment a for φ. If a is rejecting
then reject. If a is accepting then guess n+1 bits, ignore them and
accept.

If φ is not satisfiable then M(φ) will only have rejecting
paths. If φ is satisfiable then it will have at least
2n+1 accepting paths and at most 2n-1 rejecting
paths fulfilling the requirements of the definition of
BPPpath.

Han, Hemaspaandra and Thierauf prove many other results about
BPPpath. The class contains MA and
PNP[log] (P with O(log n) queries to an NP language like
SAT). BPPpath is contained in
PΣ2[log], BPPNP and
PP.

Whether BPPpath is contained in ZPPNP or Σ2 remain
the most interesting open questions about this class.

Sunday, February 16, 2003

I noticed that Slashdot has
recently mentioned some TCS research. This would therefore be
theoretical computer science that matters to nerds.

The first was a pointer to Scienceblog article
on the work of Roughgarden and Tardos on selfish routing. Here is
their main example: Consider two roads from city A to city B. The
first road takes an hour and the second road takes p hours, where p is
the fraction of people who take the second road. The best way to route
to minimize total travel time is for half the people to take the
second road (average time = 3/4 hour). If everyone acts selfishly, the
only equilibrium is everyone taking the second road for an average
time of one hour. For more details see Tim Roughgarden's home page.

The other article pointed to Microsoft's research Penny
Black Project which hopes to prevent spam by making sending email
"expensive." The
variation
using complexity requires the sender to solve a moderately hard
problem generated by the intended recipient.

Always keep in mind that what happens at Microsoft research,
especially amongst the theoreticians, should not be read to imply any
future email policy at Microsoft, no more than one can determine
future NEC policies from my own research.

Friday, February 14, 2003

Yesterday's post
on NP-complete problems reminds me of one of the more interesting open
questions, the complexity of the traveling salesman problem on the
plane.

The traveling salesman problem consists of a collection of n cities, a
symmetric distance function d(i,j) and a number k. The question is
whether there is an ordering of the cities so that a tour through
those cities and back to the start can be done in total distance at
most k. If d takes on rational values, the problem is NP-complete in
general and for many restrictions of the possible functions for d,
such as requiring d to have the triangle inequality.

Consider the case where the cities are given as points
(x,y) in the plane and the distance function is
the regular Euclidean distance, i.e.,

d((r,s),(u,v))=((r-u)2+(s-v)2)1/2.

It is open whether this traveling salesman problem on the plane is
NP-complete. In most cases, it is easy to show the problem is in NP
and more difficult to show it NP-hard. For TSP on the plane, Garey,
Graham and Johnson showed it NP-hard way back in 1976; it remains open
whether the problem is in NP.

In NP one can guess the ordering of the cities, the problem is in
checking whether the sum of distances is at most k. Since we can
approximate the square root to a polynomial number of digits the issue
is how many digits of accuracy we need. So it boils down to the
following purely mathematical question: Given an expression of length
m over integers with addition, subtraction, squares and square roots (but no
recursive squares or square roots) what is the smallest positive
number than can be expressed? If the answer is at least
2-mk for some k then TSP on the plane is in NP
and NP-complete.

Thursday, February 13, 2003

Now that we have shown that CNF-SAT is NP-complete we can use it to
show many other problems are NP-complete via the following lemma.

Lemma: If a set A is NP-complete and B is in NP and A
polynomial-time reduces to B then B is also NP-complete.

Proof: Let C be an arbitrary set in NP. We need to show C
reduces to B. We know C reduces to A (since A is complete) and A
reduces to B and it is a simple exercise to see that reductions are
transitive. ◊

For example consider 3-SAT, the set of satisfiable 3-CNF formula with
exactly 3 literals in each clause. 3-SAT is in NP by just guessing
an assignment and verifying that it satisfies the formula. To show
3-SAT is NP-complete we will reduce 3-CNF to 3-SAT. We take each
clause and convert it to a set of clauses each with three
literals. We will add additional variables for this reduction.

If the clause C has three literals then no conversion is necessary.

If the clause C has two literals something like C = x OR y. We use a new variable z and replace C with
two clauses, x OR y OR z and
x OR y OR z. Notice that C
is satisfied if and only if both new clauses can be satisfied.

If the clause C has one literal like C = x, we add two new variables,
u and v and replace C with four new clauses, x OR u OR v, x OR u OR v, x OR u OR v and x OR
u OR v.

If the clause C has k literals for k>3, we will replace C by two
clauses, one with k-1 literals and one with 3 literals and then
recurse. Suppose C = x1 OR ... OR xk. We add a
new variable w and replace C with X1 OR ... OR
Xk-2 OR w and w OR xk-1 OR
xk. Again note that C is satisfied only if both of the new
clauses are satisfied with some value for w.

That is the reduction and the proof that it works is
straightforward.

There are many many natural NP-complete problems and we could spend
many lessons showing that they are NP-complete. Personally, I find
NP-completeness proofs more algorithmic than complexity and I will
instead focus on relationships of complexity classes in future
lessons.

This site
is a collection of optimization problems but it also gives you a good
idea of what NP-complete problems are out there. The best source for
all things NP-complete still remains the excellent book
of Garey and Johnson.

Tuesday, February 11, 2003

Daniel Varga asks
about the question of NEXP not in P/poly and whether it is
"fundamentally" easier than NP not in P/poly. As a reminder: NEXP is the
class of languages accepted in nondeterministic exponential
(2nO(1)) time. P/poly are languages accepted by
nonuniform polynomial-size circuits, or equivalently by a polynomial
time machine given a polynomial amount of "advice" that depends only
on the input size.

First one must mention the
beautiful
paper
of Impagliazzo, Kabanets and Wigderson that show that
NEXP in P/poly if and only if NEXP equals MA.

NEXP not in P/poly should be much easier to prove than NP not in
P/poly, as NEXP is a much larger class than NP. Also NEXP not in
P/poly is just below the limit of what we can prove. We know that
MAEXP, the exponential time version of MA, is not contained in
P/poly. MAEXP sits just above NEXP and under some reasonable
derandomization assumptions, MAEXP = NEXP.

There is also the
issue of uniformity. If one can use the nondeterminism to reduce the
advice just a little bit than one could then diagonalize against the
P/poly machine. Also if one could slightly derandomize MA machines
than one could diagonalize NEXP from MA and thus from P/poly.

Still the problem remains difficult. BPP is contained in P/poly and we
don't even know whether BPP is different than NEXP. Virtually any weak
unconditional derandomization of BPP would separate it from NEXP but
so far we seem stuck.

Wednesday, February 05, 2003

For those with access to the Journal of the ACM, the fiftieth anniversary issue (Volume 50, Number 1)
has a number of very short articles by prominent computer scientists on a number of interesting research issues in CS. In particular, articles by Steve Cook, Juris Hartmanis,
Alexander Razborov, Peter Shor, Richard Stearns, Les Valiant and Andy Yao talk about problems directly related to this web log.

Yao's article "Classical Physics and the Church-Turing Thesis" is also available from
ECCC. He makes some arguments (that I don't necessarily agree with) that there are some physical systems where Turing machines
fail to capture computation.

Monday, February 03, 2003

The list of
accepted papers of STOC has been
posted. STOC and FOCS are the two most important annual theoretical
computer science conferences. Quite a few interesting complexity
papers. You can often download them by going to the authors' web
sites.

From Scott Aaronson: I gave a talk to my brother's high school math
club called "The Prime Facts: From Euclid to AKS." A
writeup is available in case readers of
your weblog are interested.

Last, but not least, we all are very saddened by the Columbia shuttle
tragedy and the loss of seven brave people in the pursuit of science.